Question: Determine how many solutions exist for the system of equations. ${-4x+y = 5}$ ${8x-2y = -10}$
Convert both equations to slope-intercept form: ${-4x+y = 5}$ $-4x{+4x} + y = 5{+4x}$ $y = 5+4x$ ${y = 4x+5}$ ${8x-2y = -10}$ $8x{-8x} - 2y = -10{-8x}$ $-2y = -10-8x$ $y = 5+4x$ ${y = 4x+5}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 4x+5}$ ${y = 4x+5}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-4x+y = 5}$ is also a solution of ${8x-2y = -10}$, there are infinitely many solutions.